Finite Elements in Analysis and Design 58 (2012) 44–52

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Finite Elements in Analysis and Design

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Magnetic force enhancement in a linear actuator by air-gap magnetic fielddistribution optimization and design

Jaewook Lee a,c,n, Ercan M. Dede a, Debasish Banerjee a, Hideo Iizuka a,b

a Toyota Research Institute, Toyota Motor Engineering & Manufacturing North America, Ann Arbor, MI 48105, USAb Toyota Central Research & Development Labs., Nagakute, Aichi 480 1192, Japanc Korea Aerospace University, School of Aerospace and Mechanical Engineering, Goyang-city, Geonggi-do 412-791, South Korea

a r t i c l e i n f o

Article history:

Received 11 October 2011

Received in revised form

25 March 2012

Accepted 15 April 2012Available online 10 May 2012

Keywords:

Magnetic field manipulation

Linear actuators

Magnetic forces

Structural topology optimization

4X/$ - see front matter & 2012 Elsevier B.V.

x.doi.org/10.1016/j.finel.2012.04.007

esponding author at: Korea Aerospace Univer

ical Engineering, Goyang-city, Geonggi-do 41

2 2 300 0290.

ail addresses: wookslee@kau.ac.kr, wookslee@

a b s t r a c t

The focus of this paper is to show that the magnetic force generated by a linear actuator may be

enhanced through the optimization and design of the devices air-gap magnetic field distribution.

Specifically, the use of a periodic ladder structure is proposed for magnetic field manipulation, and a

simplified finite element analysis is adopted in lieu of a higher cost computational model. The optimal

magnetic field distribution that maximizes the actuator force is then found via structural topology

optimization. This force enhancement is explained using a Maxwell stress tensor analysis and validated

through experimental studies. Finally, a periodic-ladder structure is designed for an equivalent optimal

magnetic field distribution, and the linear actuator force enhancement is confirmed.

& 2012 Elsevier B.V. All rights reserved.

1. Introduction

Modern electric vehicles are propelled by the magnetic forcegenerated within electric motors. The magnetic force in a motor isdetermined by the magnetic field present in the air-gap betweenthe stator and rotor, and a strong magnetic field ensures a largemagnetic force. This fundamental principle has led to vigorousresearch concerned with the development of strong magneticfields. For example, the optimal control of the external currentinput (i.e. magnetic field source) has previously been studied[1,2]. Additionally, Permanent Magnets (PMs) have been added tomotors as supplementary magnetic field sources [3,4]. Further-more, geometric optimization has been performed to generateeven stronger fields by reducing the motor magnetic circuitreluctance [5–8]. This prior art has focused on increasing themagnitude of the magnetic field rather than the magnetic fielddistribution, although both field strength and shape are expectedto affect the resultant magnetic force.

In a separate but related field, subwavelength techniques havebeen reported for manipulating electromagnetic fields in micro-wave and optical regimes [9–13]. Pioneering work based on aradiationless interface [14,15] has been recently extended to themanipulation of a low-frequency (kHz range) magnetic field,

All rights reserved.

sity, School of Aerospace and

2-791, South Korea.

gmail.com (J. Lee).

where the magnetic field is redistributed at some distance awayfrom a designed focusing device [16]. This manipulated magneticfield distribution may be exploited to enhance the magnetic forcein both linear actuators and motors.

Accordingly, this paper shows that magnetic force enhance-ment is possible by controlling the magnetic field distribution (asopposed to the amount of magnetic flux) using a periodic-ladderstructure positioned within a linear actuator air-gap, as shown inFig. 1(a). Following Ref. [16], the induced current in the electri-cally conductive material of the periodic-ladder structure influ-ences the air-gap magnetic field, thus the conductivity of thestructure may be specified for a desired distribution. One exampleof designed electrical conductivity, s1�s5, and the resulting air-gap magnetic field distribution are presented, respectively, in theright side image of Fig. 1(a) and solid line of Fig. 1(b). Theunmodified magnetic field distribution obtained without theperiodic-ladder structure is also presented as a dashed line inFig. 1(b). Theoretically, by designing the electrical conductivity, s,of the periodic-ladder structure, a magnetic field distribution ofany desired shape may be achieved [16].

To demonstrate magnetic force enhancement, a three-dimen-sional (3-D) finite element analysis is typically required for alinear actuator that contains a periodic-ladder structure similar tothe one shown in Fig. 1(a). During the actuator design stage, this3-D analysis is computationally expensive due to mesh refine-ment requirements for the periodic-ladder structure and air-gapregions. Thus, to efficiently find the optimal air-gap magnetic fielddistribution that maximizes the actuator magnetic force, a

Fig. 1. (a) A three-dimensional electromagnetic linear actuator model containing a periodic-ladder structure inserted within the air-gap region; (b) the resulting linear

actuator air-gap magnetic field distribution.

J. Lee et al. / Finite Elements in Analysis and Design 58 (2012) 44–52 45

simplified two-dimensional (2-D) model is proposed. The inducedcurrent in the periodic-ladder is not explicitly modeled in this 2-Danalysis. Instead, the ladder structure is replaced by a shapecontrol design region that consists of a material with high

magnetic permeability. Additionally, a PM is utilized in the 2-Dmodel instead of an electromagnet in order to fix the amount ofmagnetic flux, F, present in the system. In such a way, the shapeof this 2-D design region similarly influences the air-gap magnetic

J. Lee et al. / Finite Elements in Analysis and Design 58 (2012) 44–5246

field distribution and guides the design of the 3-D actuator withperiodic-ladder structure.

To supplement the above design process, the preferred 2-Dshape that corresponds to an optimal air-gap magnetic fielddistribution is found here using structural topology optimization[17]. Optimization results are provided to illustrate the manner inwhich a center-focused field distribution significantly increasesthe magnetic force generated by the linear actuator. This forceenhancement is then verified through experimental studies andexplained via a Maxwell stress tensor analysis. Based on theoptimal design from the simplified 2-D model, the periodic-ladderstructure is designed and force enhancement in the electromag-netic linear actuator is verified numerically through a final 3-Danalysis.

Thus, the paper is organized as follows. Section 2 describes thesimplified 2-D numerical model that is used to design theactuator air-gap magnetic field distribution. The optimal topologythat maximizes the linear actuator magnetic force is then pre-sented in Section 3. In Section 4, the optimization result isvalidated experimentally, and a physics-based interpretation ofthe force enhancement is provided. In Section 5, an equivalent3-D periodic-ladder structure is designed using the 2-D optimiza-tion result as a guide. Finally, a summary of the paper is providedin Section 6.

2. Simplified 2-D model for magnetic field manipulation

In this section, a simplified 2-D model is proposed for efficientcomputational analysis of the aforementioned electromagneticlinear actuator. A description of the model is first providedfollowed by a numerical design example.

2.1. Model description

A schematic of a basic 3-D electromagnetic linear actuatorwith a periodic-ladder structure and the assumed 2-D PM linearactuator equivalent are shown in Fig. 2 on the left and right,respectively. The model on the right eliminates out-of-planeeffects and achieves approximately the same magnetic fielddistribution through the use of a PM coupled with a materialdesign region with high magnetic permeability. Here, two sim-plifications are made: 1) shape control of the material designregion near the air-gap is assumed to mimic the effect of theperiodic-ladder structure, and 2) the magnetic source is changedfrom an electromagnet to a PM.

Regarding the first simplification, assuming a shape controlregion instead of the periodic-ladder structure is reasonablebased on the fact that magnetic fields tend to flow through highpermeability materials (e.g. cast iron), while low permeabilitymaterials (e.g. air) tend to obstruct magnetic fields. In relation tothe second simplification, for the left side model of Fig. 2 theamount of the magnetic flux, F, in the system is maintained since

Fig. 2. Schematic of a 3-D electromagnetic linear actuator with a periodic-ladder struc

constant (i.e. PM) field source plus a shape control design region (right).

the periodic-ladder structure is thin and it does not change theair-gap reluctance. This reluctance determines the magnitude ofthe magnetic field generated by the electromagnet according tomagnetic circuit theory [18]. Thus, in the right side model ofFig. 2, a PM is used to similarly maintain the amount of magneticflux during the shape control design process. By using a PMinstead of an electromagnet to energize the actuator, a constantamount of magnetic flux is generated within the system regard-less of the specific actuator shape. Thus, a PM field source plus2-D material shape design technique is used for magnetic fieldmanipulation in substitution for the modeling of a more complex3-D electromagnetic actuator with a periodic-ladder structure.

As an additional note, the magnetic field generated by the PMin the simplified model cannot be turned off. In contrast, theperiodic-ladder structure operates via an electromagnet which iscontrolled by alternating current. Hence, while not physicallypractical, the simplified model effectively replicates the magneticfield distribution of an electromagnetic actuator containing aperiodic-ladder structure.

2.2. Example of simplified model

A 2-D linear actuator model is presented in Fig. 3 that hasperformance equivalent to that of the 3-D linear actuator with aperiodic-ladder structure from Fig. 1. The designed actuator shapein Fig. 3(a) results in a magnetic field distribution, Fig. 3(b), whichis nearly identical to the field distribution obtained using theperiodic-ladder structure, Fig. 1(b).

The computational cost of the 2-D model is significantly lessthan that of the 3-D model. Specifically, the two analyses werecompared by using COMSOL Multiphysics (v.4.2) commercialfinite element software installed on a dual core workstation with3.0 GHz processors and six GB of RAM. For the 3-D analysisincluding the periodic-ladder structure shown in Fig. 1, it takes492 s to solve the problem, which has 1.41E6 degrees of freedom(DOF) from 2.21E5 second-order tetrahedral elements. The ana-lysis using the 2-D model takes only 3 s due to a significantlyreduced number of DOF (i.e. 2.3E4) from 1.08E4 second-ordertriangular elements. Note that a one-quarter symmetry modelwas used for the 3-D actuator while a one-half symmetry modelwas used for its 2-D counterpart. This example shows that the 2-Dsimplified model can reproduce the effect of the periodic-ladderstructure on the magnetic field distribution at significantly lowercomputational cost.

3. Optimization to maximize magnetic force

In this section, the optimal air-gap magnetic field distributionthat maximizes the magnetic force in a linear actuator is pre-sented. A preferred 2-D magnetic field distribution is foundthrough structural topology optimization, [17], of the actuatorstator (or yoke) close to the air-gap. First, the assumed method for

ture (left), and an equivalent simplified 2-D linear actuator model that assumes a

Fig. 3. (a) Simplified 2-D linear actuator model equivalent to the 3-D linear actuator from Fig. 1; (b) the equivalent 2-D model actuator air-gap magnetic field distribution.

J. Lee et al. / Finite Elements in Analysis and Design 58 (2012) 44–52 47

calculating the actuator magnetic force is explained. Next, theoptimization problem is formulated, and optimization results arepresented.

3.1. Magnetic force calculation

The magnetic force present in a linear actuator is calculatedusing Maxwell’s stress tensor method. Prior to applying themethod, the magnetic flux density, B, is calculated by solving

Maxwell’s equation using finite element analysis. Assuming themagnetic field is generated by a PM, there is no external currentin the electromagnetic system. Hence, Maxwell’s equation is

r � ðnr � AÞ ¼r � ðn BrÞ ðB¼r � AÞ, ð1Þ

where n is the nonlinear reluctivity, A is the magnetic vectorpotential, and Br is the residual flux density of the PM material(set to 0.5 T). Next, the magnetic force is calculated usingMaxwell’s stress tensor formulation [19], which may be

J. Lee et al. / Finite Elements in Analysis and Design 58 (2012) 44–5248

expressed as

F¼

I1

2m ðB2n�B2

t Þds

� �nþ

I1

mBnUBtds

� �t, ð2Þ

where n and t are, respectively, unit vectors normal and tangen-tial to the integration path, s, enveloping the body subject to themagnetic force.

3.2. Optimization problem

The optimal magnetic field distribution that maximizes themagnetic force acting on the plunger of a linear actuator is foundusing the simplified 2-D model proposed in Section 2. Fig. 4shows the one-half symmetry linear actuator model with thetarget design domain. The magnetic field distribution within theair-gap is optimized by finding the corresponding optimal shapewithin the design domain via structural topology optimization. Aswith conventional structural topology optimization, the densities,q, of the design domain are set as the design variables and themagnetic relative reluctivity, nr, is interpolated using the schemeproposed in Ref. [20]:

nrðrÞ ¼ nironrp1þnairð1�rp1 Þ, ð3Þ

where p1 is a penalization parameter (set to 3), nair is the relativereluctivity of air, and niron is the relative reluctivity of cast iron,which is taken as a nonlinear function of B, the magnetic fluxdensity.

niron ¼ 49:4e3:46B2

þ994:7: ð4Þ

To find the optimal density, i.e. q, distribution that maximizesthe axial-direction magnetic force, Fx, of the linear actuator, theoptimization problem is formulated as

Find q ð5Þ

Maximize FxðAÞ ð6Þ

subject to KðqÞA¼ f ð7Þ

VironðqÞ ¼ Vn

iron, ð8Þ

where K(q) and f are, respectively, the stiffness matrix and forcevector derived from the finite element formulation, Eq. (1), andreluctivity functions, Eqs. (3) and (4). Viron is the volume of thecast iron in the design domain, which is constrained to Viron

n . Theformulated optimization problem (5)–(8) is solved using aMethod of Moving Asymptotes (MMA) optimizer [21]. The sensi-tivity of the objective function, Eq. (6), for the MMA optimizer isobtained by way of the adjoint method. The reader is referred toRef. [22] for the sensitivity analysis for a structural topology

Fig. 4. One-half symmetry model of linea

optimization magnetics problem that considers nonlinearreluctivity.

3.3. Optimization result

The optimal shape and its corresponding magnetic field dis-tribution at the linear actuator air-gap are presented in Fig. 5.Observe that this optimal shape, obtained with 50% volumeconstraint (Viron

n¼0.5), focuses the magnetic field at the center

of the actuator plunger.A 50% volume constraint was chosen by comparing the

optimization object function (i.e. magnetic force) obtained usingdifferent volume constraints. As shown in Fig. 6, the magneticforce generated by the optimized structure is the greatest whenthe design domain volume is constrained to 50%. This suggeststhat the optimization result found using a 50% volume constraintis likely closest to the global optimum, while the other resultsrepresent local optimums. This is further supported by the factthat the optimization result obtained without a volume constraintalso gives a local optimum, with the resulting magnetic forcesignificantly less than that obtained with a 50% volume constraint.

4. Experimental validation and interpretation of forceenhancement.

The increase in magnetic force obtained via an optimalmagnetic field distribution was experimentally validated. Thissection provides a description of the experimental test setup aswell as measured results. A physics-based interpretation of themagnetic force enhancement is further provided using Maxwell’sstress tensor formulation, Eq. (2), from Section 3.1.

4.1. Experimental test setup

A zoomed out picture of the full linear actuator experimentaltest setup is provided in Fig. 7, for reference. The actuator yoke isconnected to ground via a non-magnetic optical breadboard, andthe plunger travels on a low friction non-magnetic linear bearing.The magnetic force acting on the plunger is reacted out throughan axial alignment coupler and threaded rod which is subse-quently connected to a ground block. The tensile force in the rodis then measured using a load cell which is positioned along theline of action of the magnetic force. The air gap between the yokeand plunger is established using a feeler gage in accordance witha preselected distance. Once the air gap is set, the force in the loadcell is measured using a digital output meter, as shown in Fig. 7.

r actuator with target design domain.

Fig. 6. The effect of the design domain volume constraint on the magnetic force

generated via the optimized topology.

Fig. 7. Linear actuator experimental test setup. Observe that the magnetic force

acting on the plunger is reacted through the alignment coupler and threaded rod

which is then attached to a ground block. The tensile force in the rod is measured

via a load cell and digital output meter.

Fig. 5. Topology optimization result using a 50% volume constraint for the design domain (left) and the corresponding linear actuator air-gap magnetic field distribution

(right).

J. Lee et al. / Finite Elements in Analysis and Design 58 (2012) 44–52 49

In the actuator experimental test setup, the magnetic force ismeasured for three variations of the linear actuator; see Models1–3 as shown, respectively, in Fig. 8 (a)–(c). A square shape designdomain was selected for Model 1 to represent a baseline unmo-dified magnetic field distribution, while the tapered designdomain in Model 2 corresponds to a straightforward centerfocused field distribution. The optimized design domain in Model3 was determined via structural topology optimization explainedin section 3 as the preferred form for controlling the magneticfield distribution and maximizing the magnetic force.

4.2. Experiment results

The experimental results for the three models, obtained usinga 0.2 mm air gap, are presented in Table 1. The measured forcevalues for Models 1 through 3 are, respectively, 7.44%, 3.48%, and47.3% less than the respective numerical force values. Thediscrepancies between the simulation and experimental results

may be attributed to three main factors. First, the exact materialproperties for both the NdFeB (i.e. PM strength) and cast iron (i.e.permeability) used in the linear actuator experimental setup wereunknown, and thus standard material properties were assumedfor all simulations. Second, for extremely focused magnetic fields,the magnitude of the field is highly sensitive to the air-gap withsmall variations leading to larger experimental error. Third, allsimulations were performed in two-dimensions, and additionalmagnetic flux leakage in the third out-of-plane dimension resultsin an end effect [23], with further reduction in magnetic force.Nonetheless, the trend of the simulation data is validated with asubstantial two-fold increase for the focused magnetic fieldpresent in the optimized, Model 3, linear actuator.

Fig. 9 presents the simulated magnetic field distribution for a0.2 mm air-gap between the linear actuator yoke and plunger inthe horizontal (i.e. x-axis) direction. For Models 2 and 3, the x-axismagnetic flux density at the y¼0 position is increased, whencompared with the distribution from Model 1, by reducing the

Table 1Comparison of simulated and measured magnetic forces plus simulated magnetic flux in the three linear actuator models from Fig. 8.

Model Model 1 Fig. 8(a) Model 2 Fig. 8(b) Model 3 Fig. 8(c)

Simulated force, Fx, on plunger (N) 80.8 (100%) 98.0 (122%) 224 (277%)

Measured force, Fx, on plunger (N) 75.2 (100%) 94.7 (126%) 152 (202%)

Magnetic flux, F, in air-gap (Wb) 1.11�10�3 (100%) 0.719�10�3 (64.6%) 1.02�10�3 (91.2%)

Fig. 9. Simulated linear actuator normal (i.e. x-axis) direction magnetic field

distributions in the air-gap between the yoke and plunger for the three models

from Fig. 8: (a) Model 1; (b) Model 2; (c) Model 3. The magnetic flux lines for the

three symmetry models are provided in the insets, for clarity.

0

20

40

60

80

100

120

140

160

180

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

Air Gap (mm)

Mag

neti

c F

orce

(N

)

Model 1

Model 3

Fig. 10. Comparison of measured force for actuator Models 1 and 3; refer to Fig. 8.

Magnetic force enhancements of 19.4% and 94.6% were measured for Model

3 relative to Model 1 for air gaps of 1 mm and 0.15 mm, respectively.

Fig. 8. Linear actuator test configurations: (a) Model 1—square shape design domain (unmodified magnetic field distribution); (b) Model 2—tapered design domain

(center focused magnetic field distribution); (c) Model 3—optimized design domain (optimally focused magnetic field distribution).

J. Lee et al. / Finite Elements in Analysis and Design 58 (2012) 44–5250

magnitude of the side field. These center concentrated fielddistributions increase the magnetic force reported for Model 1(see Table 1) by 122% and 277%, respectively, for Models 2 and 3.Observe that the magnetic force for Model 2, despite a strongerfield concentration, is less than that for Model 3. This lowermagnetic force for Model 2 is due to greater flux leakage (35.4%for Model 2 versus 8.8% for Model 3) of the magnetic field. Thus,Model 3 maximizes the actuator force by maximally exploiting thefocused field distribution while minimizing magnetic flux leakage.

To further illustrate the magnetic force enhancement over arange of air gaps, an in-depth comparison was made betweenModels 1 and 3. Fig. 10 shows the measured plunger force generated

by the linear actuator for air gaps between 0.15 and 1 mm. Observethat the magnetic force enhancement for Model 3, relative to Model1, increases for smaller air gaps. At the largest air gap, i.e. 1 mm, a19.4% increase in the magnetic force for Model 3 was measured. Thisforce enhancement for Model 3 is then substantially increased to94.6% for the smallest air gap of 0.15 mm indicating a significantshape control and magnetic field focusing effect.

4.3. Interpretation of force enhancement

An increase in magnetic force due to a focused field distribu-tion may be explained theoretically using expression (2). The

Fig. 11. Conceptual comparison of evenly distributed and focused magnetic fields.

Fig. 12. Periodic-ladder structure design for center-focused distribution.

Fig. 13. Simulated normal (i.e. x-axis) direction magnetic field distributions in the

air-gap of the linear actuator in Fig. 1(a): (a) without periodic-ladder structure;

(b) with periodic-ladder structure in Fig. 12.

J. Lee et al. / Finite Elements in Analysis and Design 58 (2012) 44–52 51

force acting on the actuator plunger is calculated by integratingthe second order terms in both the normal and tangentialdirections. The normal direction magnetic force, Fn, may bemaximized for a constant magnetic flux, F¼

R ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB2

nþB2t

qdðAreaÞ,

by focusing the distribution of Bn with zero Bt everywhere. Due tothe squared terms in the magnetic flux expression, the plungerforce resulting from a focused distribution is significantly higherthan that produced by an even distribution. The conceptualcomparison of an even versus focused magnetic field distributionis presented in Fig. 11. For the same magnetic flux, F, themagnetic force from a focused distribution is five times higherthan that from an even distribution. For the reader’s reference, thetangential force, Ft, is maximized when the focused magnetic fielddistribution is oriented 451 relative to the integration path (i.e.the amounts of Bn and Bt are identical).

5. Design of periodic-ladder structure

Based on the optimization result for the simplified 2-D linearactuator model from Section 3, the periodic-ladder structure forthe full 3-D linear actuator was designed. Specifically, in order toachieve a center-focused air-gap magnetic field distribution, theelectric conductivity, s, of the periodic-ladder structure wascontrolled, as shown in Fig. 12. A periodic-ladder comprising 10unit cells was assumed for the design, where the four outer unitcells on each side of the structure centerline were found to havean electrical conductivity equal to that of copper. The two inner

unit cells of the ladder were then assigned electrical conductivityvalues equal to zero corresponding to a dielectric material.

The resulting center-focused air-gap magnetic field distribu-tion dictated by the periodic-ladder structure, and found using a3-D simulation of the linear actuator, is presented in Fig. 13. Forthe reference, the distribution without the periodic-ladder struc-ture is also presented. The 3-D simulation predicts a considerablemagnetic force increase due to the focused field distribution. Theforce with the periodic-ladder structure was calculated as 14.4 N,which is 55.8% higher than the force without the periodic-ladderstructure (9.21 N). These results were obtained using an externalcoil that was excited with a 5 Amp, 10 kHz AC signal.

6. Discussion and conclusion

The magnetic force in a linear actuator was enhanced bymanipulating the air-gap magnetic field distribution using aperiodic-ladder structure with designed electrical conductivity. Adesign method was proposed, where a 2-D computational modelwas exploited in designing a 3-D actuator system. This simplifiedmodeling approach was shown to be effective in reproducing thedesired magnetic field focusing effect. Additionally, the magneticfield distribution was designed in 2-D to maximize the magneticforce using structural topology optimization. Subsequent experi-mental investigations confirmed a significant two-fold increase inthe linear actuator force due to an optimally focused magneticfield distribution. Finally, a 3-D periodic-ladder structure wassynthesized for an equivalent optimal magnetic field distribution,and an increase of 55.8% in the linear actuator force was predictedusing the designed structure.

As a final point, the predicted force increase using magneticfield focusing might be limited if the magnetic field is severelysaturated within the focused region. In practical motor or actua-tor applications, the magnetic field may be saturated under someoperating conditions. For these conditions, magnetic field shield-ing or diffusion might be more effective than field focusing in

J. Lee et al. / Finite Elements in Analysis and Design 58 (2012) 44–5252

order to improve the system performance. To find the optimaldesign for shielding or diffusion, the proposed analysis and designmethod is still useful and its extension is an interesting topic forfuture research. Thus, a well designed periodic-ladder structuremay be used in a variety of beneficial ways to manipulate themagnetic field. Accordingly, the proper selection of a magneticfield manipulation technique is central to the advanced develop-ment of efficient linear actuators and motors for modern electro-mechanical vehicle systems.

Acknowledgment

We acknowledge fruitful discussions with Dr. Mindy Zhang,Dr. Michael Rowe, and Dr. Joy Wu.

References

[1] S.L. Herman, Industrial Motor Control, sixth ed., Delmar, New York, 2010.[2] A. Hughes, Electric Motors and Drives, third ed., Elsevier, Massachusetts,

2006.[3] D.C. Hanselman, Brushless Permanent Magnet Motor Design, 2nd ed., Magna

Physics Publishing, Ohio, 2006.[4] D.R. Krishnan, Permanent Magnet Synchronous and Brushless DC Motor

Drives, CRC Press, New York, 2010.[5] J.S. Choi, J. Yoo, Simultaneous structural topology optimization of electro-

magnetic sources and ferromagnetic materials, Comput. Methods Appl. Mech.Eng. 198 (2009) 2111–2121.

[6] J. Lee, N. Kikuchi, Structural topology optimization of electrical machinery tomaximize stiffness with body force distribution, IEEE Trans. Magn. 46 (2010)3790–3794.

[7] S.I. Park, S. Min, Magnetic actuator design for maximizing force using level setbased topology optimization, IEEE Trans. Magn. 45 (2009) 2336–2339.

[8] J. Yoo, S. Yang, J.S. Choi, Optimal design of an electromagnetic coupler tomaximize force to a specific direction, IEEE Trans. Magn. 44 (2008)1737–1742.

[9] K.R. Chen, Focusing of light beyond the diffraction limit of half the wave-length, Opt. Lett. 35 (2010) 3763–3765.

[10] Y. Fu, W. Zhou, L.E.N. Lim, C.L. Du, X.G. Luo, Plasmonic micro-zone plate:superfocusing at visible regime, Appl. Phys. Lett. 91 (2007) 061124.

[11] G. Lerosey, J. Rosny, A. Tourin, M. Fink, Focusing beyond diffraction limit withfar-field time reversal, Science 315 (2007) 1120–1122.

[12] L. Markley, A.M.H. Wong, Y. Wang, G.V. Eleftheriades, Spatially shifted beamapproach to sub-wavelength focusing, Phys. Rev. Lett. 101 (2008) 113901.

[13] L. Verslegers, P.B. Catrysse, Z. Yu, S. Fan, Deep-subwavelength focusing andsteering of light in an aperiodic metallic waveguide array, Phys. Rev. Lett. 103(2009) 033902.

[14] A. Grbic, L. Jiang, R. Merlin, Near-field plates: subdiffraction focusing withpatterned surfaces, Science 320 (2008) 511–513.

[15] R. Merlin, Radiationless electromagnetic interference: evanescent-fieldlenses and perfect focusing, Science 317 (2007) 927–929.

[16] D. Banerjee, J. Lee, E.M. Dede, H. Iizuka, Kilohertz magnetic field focusing in apair of metallic periodic-ladder structures, Appl. Phys. Lett. 99 (2011)093501.

[17] M.P. Bendsøe, O. Sigmund, Topology Optimization, 2nd ed., Springer, NewYork, 2003.

[18] D. Zorbas, Electric Machines: Principles, Applications, and Control Sche-matics, West Pub. Co., 1989 Technology and Engineering.

[19] M. Ito, M.F. Tajima, H. Kanazawa, Evaluation of force calculating methods,IEEE Trans. Magn. 26 (2010) 1035–1038.

[20] J.S. Choi, J. Yoo, Structural optimization of ferromagnetic materials based onthe magnetic reluctivity for magnetic field problems, Comput. Methods Appl.Mech. Eng. 197 (2009) 4193–4206.

[21] K. Svanberg, The method of moving asymptotes—a new method for struc-tural optimization, Int. J. Numer. Methods Eng. 24 (1987) 359–373.

[22] S. Park, S. Min, Design of magnetic actuator with nonlinear ferromagneticmaterials using level-set based topology optimization, IEEE Trans. Magn. 46(2010) 618–621.

[23] W. Wu, J.B. Dunlop, S.J. Collocott, B.A. Kalan, Design optimization of aswitched reluctance motor by electromagnetic and thermal finite-elementanalysis, IEEE Trans. Magn. 39 (2003) 3334–3336.